发表机构
University of Pisa; Saarland University; University of Amsterdam（比萨大学; 萨尔兰大学; 阿姆斯特丹大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出SPEED-AE框架，结合因果表示学习与自编码器，从图像等非结构化数据中恢复稀疏多项式ODE，在多个动力系统上提升解缠性与预测精度。
AI 中文摘要
我们研究了从非结构化、高维观测（如图像）中恢复动力系统控制常微分方程（ODE）的问题。现有的ODE发现方法通常假设对变量进行直接测量，或者对学习到的变量和方程不提供理论保证。虽然因果表示学习（CRL）方法提供了在高维观测中识别变量直至分量级微分同胚的保证，但我们表明，在一般情况下，这些变量不能直接用作方程发现方法的输入，因为方程发现方法通常假设变量将导致稀疏方程。因此，我们引入了稀疏等价方程发现自编码器（SPEED-AE），这是一个将预训练的CRL方法与分量级自编码器相结合的框架，该自编码器学习适合稀疏ODE发现的变量变换。我们证明，对于多项式ODE，这一额外步骤使我们能够将每个变量的可识别性从多项式微分同胚限制为单项式微分同胚。在Lotka-Volterra、Lorenz和双摆系统上的实验表明，SPEED-AE改进了CRL方法的解缠性，并恢复了最接近真实情况的ODE，同时实现了最先进的预测性能。
英文摘要
We study the problem of recovering the governing ODE of a dynamical system from unstructured, high-dimensional observations such as images. Existing methods for ODE discovery typically assume direct measurements of the variables, or do not provide theoretical guarantees on the learned variables and equations. While Causal Representation Learning (CRL) methods provide guarantees on identifying variables from high-dimensional observations up to component-wise diffeomorphisms, we show that in general these variables cannot be used directly as input to equation discovery methods, which typically assume that the variables will lead to sparse equations. So we introduce SParse Equivalent Equation Discovery AutoEncoder (SPEED-AE), a framework that combines a pretrained CRL method with a component-wise autoencoder that learns transformations of variables that are amenable to sparse ODE discovery. We show that for polynomial ODEs, this additional step allows us to restrict the identifiability of each variable from polynomial to monomial diffeomorphisms. Experiments on Lotka-Volterra, Lorenz, and a two-pendulum system show that SPEED-AE improves on the disentanglement of the CRL methods and that it recovers ODEs that are closest to the ground truth, while achieving state-of-the-art forecasting performance.
